Steady-state queue size distribution of discrete-time PH/Geo/1 retrial queues

This paper studies discrete-time single server retrial queues with phase-type inter-arrival times and geometric service times. A matrix analytical method is applied to derive the analytical solution for the joint steady-state distribution of arrival phases and queue sizes of the system. The necessary and sufficient condition for system stability is also determined. Based on the special form of the matrix analytic solution of the system, upper and lower bounds for the joint steady-state distribution of arrival phases and queue sizes are developed. It is shown that the errors between these upper and lower bounds can be made as small as desired. Finally, an efficient and numerically stable algorithm for computing the joint steady-state probabilities of arrival phases and queue sizes is presented.